Tropical Varieties for Exponential Sums and Their Distance to Amoebae

Abstract. Given any n-variate exponential sum, g, the real part of the complex zero set of g forms a sub-analytic variety R(Z(g)) generalizing the amoeba of a complex polynomial. We extend the notion of Archimedean tropical hypersurface to derive a piecewise linear approximation, Trop(g), of R(Z(g)), with explicit bounds — solely as a function of n, the number of terms, and the minimal distance between frequencies — for the Hausdorff distance ∆(R(Z(g)),Trop(g)). We also discuss the membership complexity of Trop(g) relative to the Blum-Shub-Smale computational model over R. Along the way, we also estimate the number of roots of univariate exponential sums in axis-parallel rectangles, refining earlier work of Wilder and Voorhoeve.